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        <td class="header">&nbsp; Polarimetric Covariance or Coherency Matrices Generation</td>
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<h3>Polarimetric Covariance or Coherency Matrices Generation Operator</h3>&nbsp;&nbsp;&nbsp;This
operator creates the following polarimetric covariance or coherency
matrices for a given full polarimetric SAR product:<br>
<ul>
    <li>covariance matrix C3</li>
    <li>covariance matrix C4</li>
    <li>coherency matrix T3</li>
    <li>coherency matrix T4</li>
</ul>
<h4>Covariance Matrix C<sub>4</sub></h4>&nbsp;&nbsp; Let <br>

<div style="margin-left: 40px;"><img style="width: 127px; height: 63px;" alt=""
                                     src="images/covarianceMatrixGenerationOp_eq1.jpeg"><br></div>
<br>&nbsp;&nbsp;&nbsp; be the complex Sinclair scatter matrix and<br>

<div style="margin-left: 40px;"><img style="width: 230px; height: 33px;" alt=""
                                     src="images/covarianceMatrixGenerationOp_eq2.jpeg"><br></div>
<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span
        style="position: relative; top: 6pt;"></span></span><span
        style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span
        style="position: relative; top: 6pt;"></span></span>&nbsp;<br>&nbsp;&nbsp;&nbsp;
be the 4-D target vector, where superscript&nbsp;T stands for the
transpose operator. Then the (4x4) covariance matrix&nbsp;C<sub>4</sub> is defined as<br>

<div style="margin-left: 40px;">&nbsp;&nbsp; <img style="width: 80px; height: 30px;" alt=""
                                                  src="images/covarianceMatrixGenerationOp_eq3.jpeg"><br></div>
&nbsp;&nbsp;&nbsp; where superscript H represents the transpose conjugate operator.<br><h4>Covariance Matrix
    C<sub>3</sub></h4>&nbsp;&nbsp;
For monostatic backscattering case, the transmitter and the receiver
are collocated. The reciprocity constrains the Sinclair scattering
matrix to be symmetrical, i.e. S<sub>hv</sub> = S<sub>vh</sub>. The 3-D target vector becomes<br>

<div style="margin-left: 40px;"><img style="width: 203px; height: 38px;" alt=""
                                     src="images/covarianceMatrixGenerationOp_eq4.jpeg"><br></div>
<span style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span
        style="position: relative; top: 6pt;"></span></span><span
        style="font-size: 12pt; font-family: &quot;Times New Roman&quot;;"><span
        style="position: relative; top: 6pt;"></span></span>&nbsp;<br>&nbsp;&nbsp;&nbsp;&nbsp;Then the (3x3) covariance
matrix&nbsp;C<sub>3</sub> is given by<br>

<div style="margin-left: 40px;"><img style="width: 78px; height: 32px;" alt=""
                                     src="images/covarianceMatrixGenerationOp_eq5.jpeg"><br></div>
<h4>Cohrency Matrix T<sub>4</sub></h4>&nbsp;&nbsp; Let &nbsp;the 4-D target vector be defined as the follows<br>

<div style="margin-left: 40px;"><img style="width: 462px; height: 55px;" alt=""
                                     src="images/coherencyMatrixGenerationOp_eq1.jpeg"><br></div>
&nbsp;&nbsp;&nbsp;&nbsp;Then the (4x4) coherency matrix T<sub>4</sub> is given by<br>

<div style="margin-left: 40px;"><img style="width: 77px; height: 30px;" alt=""
                                     src="images/coherencyMatrixGenerationOp_eq2.jpeg"></div>
<h4>Cohrency Matrix T<sub>3</sub></h4>&nbsp;&nbsp; For monostatic backscattering case,&nbsp;&nbsp;the&nbsp;target vector
becomes<br>

<div style="margin-left: 40px;"><img style="width: 312px; height: 55px;" alt=""
                                     src="images/coherencyMatrixGenerationOp_eq3.jpeg"><br></div>
&nbsp;&nbsp;&nbsp;&nbsp;Then the (3x3) coherency matrix T<sub>3</sub> is given by<br>

<div style="margin-left: 40px;"><img style="width: 77px; height: 32px;" alt=""
                                     src="images/coherencyMatrixGenerationOp_eq4.jpeg"><br></div>
<h4>Input and Output</h4>
<ul>
    <li>The
        input to this operator is a full polarimetric SAR product with 8 bands,
        i.e. I and Q bands for HH, VV, HV and VH polarizations.
    </li>
    <li>Since the output covariance or coherency matrix is Hermitian positive semidefinite, only 9 elements
        in&nbsp;C<sub>3</sub> or T<sub>3</sub>&nbsp;are independent, and 16 elements in C<sub>4</sub> or T<sub>4</sub>
        are independent. Therefore only the independent elements are output. For example, the following 9 bands are
        output:&nbsp;C<sub>11</sub>,&nbsp;C<sub>22_real</sub>, C<sub>12_imag</sub>, C<sub>13_real</sub>,
        C<sub>13_imag</sub>, C<sub>22</sub>, C<sub>23_real</sub>, C<sub>23_imag</sub>, C<sub>33</sub> for covariance
        matrix C<sub>3</sub>.
    </li>
</ul>
<ol>
</ol>
<h4>Parameters Used</h4>
<ol>
    <li>&nbsp;&nbsp; Polarimetrix Matrix: The covariance or coherency matrix type. The available types
        are:&nbsp;C<sub>3</sub>,&nbsp;C<sub>4</sub>, T<sub>3</sub> and T<sub>4</sub>.
    </li>
</ol>
<br>
<img style="width: 500px; height: 500px;" alt="" src="images/coherencyMatrixGenerationOp.jpeg"><br>

<p> Reference:&nbsp;</p>

<p>[1] Jong-Sen Lee and Eric Pottier, Polarimetric Radar Imaging: From Basics to Applications, CRC Press, 2009</p>
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